Elsewhere in Des Moines at the same time, Mr Trump led a raucous rally in honour of the country's war veterans.

"When you're treated badly, you have to stick up for your rights," he said, referring to his row with Fox.

Analysis by Anthony Zurcher, BBC News, Des Moines

So this is what a Republican debate would have looked like in an alternate universe, where Donald Trump was still a reality show celebrity and a failed steak salesman.

With a few notable exceptions, all the candidates kept their balance and stayed on their talking points. It's as if the unconventional Trump's absence from the hall made everyone a bit more restrained and, well, political.

The broadcaster released a statement that said Mr Trump offered to appear at the debate if Fox contributed $5m (£3.4m) to his charities, but they refused.

Data released by Google after the debate suggested that search interest in Mr Trump still far surpassed the other candidates.

Many observers on social media thought the event was duller without the brash New Yorker.

Image copyrightTwitter

But others remarked how the absence of his dominating personality helped other candidates to blossom.

Image copyrightTwitter

Some of the night's most heated moments came during exchanges about immigration.

Mr Cruz and Mr Rubio were both forced to explain video clips of previous statements that appeared to be at odds with their hardline campaign pledges.

They then turned on each other, with Mr Rubio saying immigration was "the lie that Ted's campaign is built upon".

An Iraq war veteran who came to the US from Mexico as a child appeared via YouTube to tell the candidates that "some of the comments in this campaign make us question our place in this country".

Mr Bush applauded Dulce Candy and said "we should be a welcoming nation".

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Media captionThree things Trump says

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Media captionThree things Cruz says

The Iowa caucuses on Monday are seen as the first real test of the election campaign, and the beginning of a series of state-by-state contests to choose delegates for both Republicans and Democrats.

Unlike a primary, which is a traditional election featuring secret ballots on polling day, the caucuses in Iowa are meetings of registered party voters and activists where they discuss the candidates and then vote.

'Gone but not forgotten'- media reaction

Many US newspaper commentators appear to agree that although the Iowa debate was "Trump-free", he was still a big presence.

"It turns out that Donald Trump can dominate a debate even if he doesn't show up on stage," wrote Susan Page in USA Today.

"The evening offered an almost perfect reflection of the state of the Republican race, with Trump occupying his own space and the rest of the candidates competing with one another to emerge as his principal rival. In his absence, the other hopefuls struggled with only limited success to distinguish themselves as the best of the rest," said Dan Balz in the Washington Post.

"The debate gave Iowa voters a chance to see all seven men one last time before the caucuses. With a good share of Republicans still undecided or open to changing candidates, the lack of scene-stealing by Mr Trump… could pay some dividends for the candidates here or in New Hampshire , where many of them are competing for second place after Mr Trump," said Patrick Healy and Jonathan Martin in the New York Times.

"Trump was gone but not forgotten," said Mark Z Barabak, Michael Finnegan and Seema Mehta in the LA Times. But they said the debate "seemed unlikely to dramatically recast the race ahead of Monday's Iowa caucuses".